This is cool. Instead of thinking of a linear fractional transform as a continuous function that traces out a hyperbola, or as an interval on the real number line, we can think of a linear fractional transform as a function that computes a permutation.

Here is a function called d3 that is defined on the symbols a, b, c, d, e, and f.

1 ]=> (d3 'a 'b)
;Value: d

With six symbols, there's only 36 possible outcomes, so we can enumerate them: